Few-fermion thermometry Phys. Rev. A 97, 063619 (2018) Tomasz Sowiński Institute of Physics of the Polish Academy of Sciences Co-authors: Marcin Płodzień Rafał Demkowicz-Dobrzański FEW-BODY PROBLEMS FewBody.ifpan.edu.pl
motivation Selim Jochim s experimental group in Heidelberg 6Li Atomic number (Z) 3 Nucleons (Z+N) 6 Total electronic spin (S) 1/2 Total nuclear spin (I) 1 message from experiments two flavours of spinless fermions well controlled number of particles one-dimensional confinement strong interactions between flavours Hyperfine states (F=S+I) 1/2 or 3/2 "i = F =3/2,m F = 3/2i #i = F =1/2,m F =1/2i what for??? extreme control on the system extreme precision of measurements from one' to 'many' crossover
message from experiments F. Serwane et al., Science 332, 6027 (2011)
message from experiments F. Serwane et al., Science 332, 6027 (2011)
the model two distinguishable flavours of fermions ( and ) both flavours may have different masses both flavours confined in one-dimensional trap opposite spins do interact via sort range δ-like potential Ĥ = X Z dx ˆ (x) + g Z apple ~ 2 2m d 2 dx 2 + V (x) dx ˆ # (x) ˆ " (x) ˆ "(x) ˆ #(x) ˆ (x) (anti-)commutation relations n ˆ (x), ˆ (x )o 0 = (x x 0 ) n ˆ (x), ˆ (x )o 0 =0 h Ĥi h Ĥi ˆN", = ˆN#, =0 Z ˆN = dx ˆ (x) ˆ (x)
many-body spectrum Notice the quasi-degeneracy of the spectrum for strong repulsions! Utilise it for precise thermometry!
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sha1_base64="wfdgubqrdcqbckq9yt5kqmegvzq=">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</latexit> thermal sensitivity ˆ T = 1 Z e Ĥ/k BT = X i P i E i ihe i Measurements: Some set of probabilities as functions of T. The limit is determined by the Fisher information. {p i (T )}! F({p i })= X i T 2 1 n F({p i }) 1 p i dpi dt 2 IF: One can measure Gibbs probabilities {p i (T )} = {P i (T )} then the Fisher information is the largest possible! F Q (ˆ T )= Ĥ2 T 4 = Tr[ˆ T Ĥ2 ] Tr[ˆ T Ĥ] 2 T 4
quantum Fisher information Fisher Information 10 3 10 2 N = N = 2 g=14 F Q (ˆ T )= 10 1 0 0.02 0.04 0.06 0.08 F Q (ˆ T )= Temperature T Ĥ2 T 4 = Tr[ˆ T Ĥ2 ] Tr[ˆ T Ĥ] 2 T 4
the simplest scenario Fisher Information 10 3 10 2 N = N = 2 g=14 F Q (ˆ T )= 10 1 0 0.02 0.04 0.06 0.08 Temperature T The probability of finding a particle just above the Fermi level
Occupations of the lowest orbitals Fisher Information 10 3 10 2 N = N = 2 g=14 F Q (ˆ T )= 10 1 0 0.02 0.04 0.06 0.08 Temperature T Probabilities p 0, p 1, p 2, p 3, p 4 that in two the lowest orbitals one can find 0, 1, 2, 3, or 4 particles. {
All occupations are known Fisher Information 10 3 10 2 N = N = 2 g=14 F Q (ˆ T )= 10 1 0 0.02 0.04 0.06 0.08 Temperature T When we know all probabilities of finding four particles in different configurations
the fundamental bound Quantum Fisher Information 10 4 10 3 10 2 10 1 g = 14 g = 10 Fundamental bound g = 5 Fundamental bound when level spacings can be engineered N = N = 2 L. A. Correa et al. Phys. Rev. Lett. 114, 220405 (2015). 1 10-1 g = 0 10-3 10-2 10-1 1 Temperature T
Few-fermion thermometry Phys. Rev. A 97, 063619 (2018) Tomasz Sowiński Institute of Physics of the Polish Academy of Sciences Co-authors: Marcin Płodzień Rafał Demkowicz-Dobrzański FEW-BODY PROBLEMS FewBody.ifpan.edu.pl